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Math Help - Finding 2 vertices of rectangle with 2 known vertices

  1. #1
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    Finding 2 vertices of rectangle with 2 known vertices

    Rectangle ABCD
    A is (-1,4), B is (4,8)
    Equation of BC is 4y + 5x= 52
    equation of AD is 4y + 5x = 11
    equation of AB is 5y= 4x+ 24

    Find C and D
    Answer is (12,-2) and (7,-6)

    I've tried gradient,length,midpoint formulas but to no avail.
    Thanks
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  2. #2
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    Re: Finding 2 vertices of rectangle with 2 known vertices

    The problem, as posed, has no unique solution. Is there some other attribute of the rectangle which you have not posted, such as its area?
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    Re: Finding 2 vertices of rectangle with 2 known vertices

    Apologies..I left out
    Length of BC is twice the length Of AB
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  4. #4
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    Re: Finding 2 vertices of rectangle with 2 known vertices

    Quote Originally Posted by Rodimus79 View Post
    Rectangle ABCD
    A is (-1,4), B is (4,8)
    Equation of BC is 4y + 5x= 52
    equation of AD is 4y + 5x = 11
    equation of AB is 5y= 4x+ 24
    And, you now tell us, "Length of BC is twice the length Of AB"
    It is easy to calculate that the length of AB is \sqrt{(4-(-1)^2+ (8- 4)^2}= \sqrt{25+ 16}= \sqrt{41}
    so the length of BC is 2\sqrt{41} and, because this is a rectangle, BC and AD are the same length.

    Take D= (x, y)= (x, (11- 5x)/4). Then we must have [tex]\sqrt{(x- (-1))^2+ ((11- 5x)/4- 4)^2}= 2\sqrt{41}[tex]
    so that, squaring both sides gives (x+1)^2+ (5x+ 5)^2/16= 164. Solve that quadratic equation for x. Do the same with BC.

    Find C and D
    Answer is (12,-2) and (7,-6)

    I've tried gradient,length,midpoint formulas but to no avail.
    Thanks
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    Re: Finding 2 vertices of rectangle with 2 known vertices

    Thanks so much, HallsofIvy!! The key step was to have y = 11-5x/4, to express y in terms of x, since it's a point on the line and satisifies the equation! Thanks once again!!!
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    Re: Finding 2 vertices of rectangle with 2 known vertices

    Finding 2 vertices of rectangle with 2 known vertices-15-aug-13.png
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    Re: Finding 2 vertices of rectangle with 2 known vertices

    given the two points and the fact that the length of the rectangle is twice width(AB) there is a simple method to find the other vertices
    slope diagram AB = 4/5
    slope diagram AD =-5/4
    actual slope diagram AD =-10/8 since AD =2AC
    Point D x =8-1 =7 y =4 + absolute value y =10 y=-6
    same procedure gives point C
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  8. #8
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    Re: Finding 2 vertices of rectangle with 2 known vertices

    Quote Originally Posted by bjhopper View Post
    given the two points and the fact that the length of the rectangle is twice width(AB) there is a simple method to find the other vertices
    slope diagram AB = 4/5
    slope diagram AD =-5/4
    actual slope diagram AD =-10/8 since AD =2AC
    Point D x =8-1 =7 y =4 + absolute value y =10 y=-6
    same procedure gives point C
    The slope has nothing to do with the length of line segment? thus the slope of Ad is -5/4
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  9. #9
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    Re: Finding 2 vertices of rectangle with 2 known vertices

    Quote Originally Posted by ibdutt View Post
    The slope has nothing to do with the length of line segment? thus the slope of Ad is -5/4

    Actual slope of AD = -10/8. This slope defines the length of AD which is not used in my solution
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